does having components of my invention as prior art hinder my ability to get a patent on my invention?
Having prior art usually makes things more difficult, but doesn't necessarily preclude you from obtaining a patent. It really depends on what are in the prior art's claims. In general, you don't just claim component A, you claim component A as applied in a specific situation. Lets say someone has a patent for compound A to treat cancer and someone else has compound B to treat cancer, you might run up against an rejection due to "obviousness" if you are try to patent the combination of A and B to treat cancer. Obviousness is where the examiner rejects the application because she feels that the combination is something someone with ordinary skill in the field would try. Just answering "well, no one else tried it so it must not be obvious" likely won't be convincing. In my experience, if there is an unexpected benefit, you have a better chance of avoiding an obviousness rejection. Lets say A and B have 1X effects, but the combination of A and B have a 10X effect. It is also helpful if say A is applied to one problem and B is applied to a different problem. Then the combination would seem less likely to work in concert.
Is there a way to write my claim that doesn’t infringe the other patents?
I am not a patent attorney. I have more than 80 patents worldwide, but would never attempt to draft my own claims. I have sometimes asked my attorneys why a claim’s language seems so arcane, and have gotten a response referring to this or that legal precedent. That said, what is written in the specification has a real impact to what can be claimed. In my experience, if you are aware of important prior art that might be used against you, it is better to be proactive and explain in the background section of the application why you are different and why the combinations you are inventive are non-obvious. Remember, claims can be amended, but the specification can't.
I should point out that if some of the components you use in combination are applied to the same problem, you may need to obtain a license to use your invention. If, for instance A is applied to cancer and B is applied to diabetes. If your invention uses A and B to treat cancer, you probably need a license to use A. However if you want to apply A or B or the combination to treat a different disease, you might avoid needing licenses.